Stochastic Epidemic Models and Their Statistical Analysis

139 Pages·2000·3.26 MB·English

by  Håkan Andersson

Lecture Notes in Statistics 151
Edited by P. Bickel, P. Diggle, S. Fienberg, K. Krickeberg,
I. Olkin, N. Wermuth, S. Zeger
Springer Science+Business Media, LLC
Andersson
Hăkan
Tom Britton
Stochastic Epidemie Models and
Their Statistical Analysis
t
Springer
Hăkan Andersson Tom Britton
Group Financial Risk Control Department ofMathematics
SwedBank Uppsala University
SE-105 34 Stockholm PO Box 480
Sweden SE-751 06 Uppsala
[email protected] Sweden
[email protected]
Library of Congress Cataloging-in-Publication Data
8tochastic epidemie models and their statistical analysis / Hâkan Andersson, Tom Britton.
p. cm. -- (Lecture notes in statistics; 151)
Inc1udes bibliographical references and index.
I8BN 978-0-387-95050-1 I8BN 978-1-4612-1158-7 (eBook)
DOI 10.1007/978-1-4612-1158-7
1. Epidemiology--Mathematical models. 2. Epidemiology--8tatistical methods. 3.
8tochastic analysis. L Andersson, Hâkan. II. Britton, Tom. III. Lec!ure notes in statistics
(8pringer-Verlag); v. 151.
RA652.2.M3 8762000
614.4'01'5118--dc21
00-041911
Printed on acid-free paper.
© 2000 Springer Science+Business Media New York
Originally published by Springer-Verlag New York, Inc. in 2000
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9 876 5 4 321
ISBN 978-0-387-95050-1
Preface
The present lecture notes describe stochastic epidemic models and methods for their
statistical analysis. Our aim is to present ideas for such models, and methods for their
analysis; along the way we make practical use of several probabilistic and statistical
techniques. This will be done without focusing on any specific disease, and instead
rigorously analyzing rather simple models. The reader of these lecture notes could
thus have a two-fold purpose in mind: to learn about epidemic models and their
statistical analysis, and/or to learn and apply techniques in probability and statistics.
The lecture notes require an early graduate level knowledge of probability and
statistics. They introduce several techniques which might be new to students, but our
intention is to present these keeping the technical level at a minlmum. Techniques that
are explained and applied in the lecture notes are, for example: coupling, diffusion
approximation, random graphs, likelihood theory for counting processes, martingales,
the EM-algorithm and MCMC methods. The aim is to introduce and apply these
techniques, thus hopefully motivating their further theoretical treatment. A few
sections, mainly in Chapter 5, assume some knowledge of weak convergence; we hope
that readers not familiar with this theory can understand the these parts at a heuristic
level.
The text is divided into two distinct but related parts: modelling and estimation.
The first part covers stochastic models and their properties, often assuming a large
community in which the disease is spread. The second part deals with statistical
questions, that is, what can be said about the model and its parameters, given that
an epidemic outbreak has been observed. The second part uses results from the first
part, and is hence not suited for reading without having read the first part.
The lecture notes are self-instructive and may be read by anyone interested in
the area. They are suited for a one-semester course of approximately 15 two-hour
lectures. Most chapters may be presented in one such lecture. Chapters that need
somewhat longer treatment are Chapters 5, 6, and 8. Each chapter ends with a few
exercises giving extensions of the theory presented in the text..
These notes were written during the spring term 1999 when the authors gave a
joint graduate course in the Departments of Mathematics at Stockholm and Uppsala
Universities. We thank the participants in the course: Anders Bjorkstrom, Nestor
Correia, Maria Deijfen, Peter Grenholm, Annika Gunnerhed, Allan Gut, Jemila Seid
VI Preface
Hamid, Stefan Israelsson, Kristi Kuljus, Johan Lindback and Habte Tewoldeberhan
for their constructive criticisms of the manuscript as well as for pointing out several
errors. The lecture notes were re-written taking their comments into account during
the second half 1999. We are also grateful to the referees of Springer for careful
reading and numerous constructive suggestions on how to clearify bits and pieces as
well as language improvement. Needless to say, the authors are responsible for any
remaining errors. Tom Britton gratefully acknowledges support from the Swedish
Natural Science Research Council.
Hdkan Andersson, Stockholm February, 2000
Tom Britton, Uppsala
Contents
Preface v
Part I: STOCHASTIC MODELLING 1
Chapter 1. Introduction 3
1.1. Stochastic versus deterministic models 3
1.2. A simple epidemic model: The Reed-Frost model 4
1.3. Stochastic epidemics in large communities 6
1.4. History of epidemic modelling 7
Exercises 9
Chapter 2. The standard SIR epidemic model 11
2.1. Definition of the model 11
2.2. The Sellke construction 12
2.3. The Markovian case 14
2.4. Exact results 15
Exercises 18
Chapter 3. Coupling methods 19
3.1. First examples 19
3.2. Definition of coupling 22
3.3. Applications to epidemics 22
Exercises 26
Chapter 4. The threshold limit theorem 27
4.1. The imbedded process 27
4.2. Preliminary convergence results 28
Vlll Contents
4.3. The case mn/n -+ J1 > 0 as n -+ 00 30
4.4. The case mn = m for all n 32
4.5. Duration of the Markovian SIR epidemic 34
Exercises 36
Chapter 5. Density dependent jump Markov processes 39
5.1. An example: A simple birth and death process 39
5.2. The general model 40
5.3. The Law of Large Numbers 41
5.4. The Central Limit Theorem 43
5.5. Applications to epidemic models 46
Exercises 48
Chapter 6. Multitype epidemics 51
6.1. The standard SIR multitype epidemic model 51
6.2. Large population limits 53
6.3. Household model 55
6.4. Comparing equal and varying susceptibility 56
Exercises 61
Chapter 7. Epidemics and graphs 63
7.1. Random graph interpretation 64
7.2. Constant infectious period 65
7.3. Epidemics and social networks 66
7.4. The two-dimensional lattice 70
Exercises 72
Chapter 8. Models for endemic diseases 73
8.1. The SIR model with demography 73
8.2. The SIS model 77
Exercises 83
Contents IX
Part II: ESTIMATION 85
Chapter 9. Complete observation of the epidemic process 87
9.1. Martingales and log-likelihoods of counting processes 87
9.2. ML-estimation for the standard SIR epidemic 91
Exercises 94
Chapter 10. Estimation in partially observed epidemics 99
10.1. Estimation based on martingale methods 99
10.2. Estimation based on the EM-algorithm 103
Exercises 105
Chapter 11. Markov Chain Monte Carlo methods 107
11.1. Description of the techniques 107
ll.2. Important examples 109
11.3. Practical implementation issues III
11.4. Bayesian inference for epidemics ll3
Exercises 114
Chapter 12. Vaccination 117
12.l. Estimating vaccination policies based on one epidemic ll7
12.2. Estimating vaccination policies for endemic diseases 120
12.3. Estimation of vaccine efficacy 123
Exercises 124
References 127
Subject index 135
Part I
STOCHASTIC MODELLING
In the first part of these lecture notes, we present stochastic models for the spread
of an infectious disease in some given population. We stress that the models aim at
describing the spread of viral or bacterial infections with a person-to-person trans
mission mechanism. Diseases that belong to this category are, for example, childhood
diseases (measles, chickenpox, mumps, rubella, ... ), STD's (sexually transmitted dis
eases) and less severe diseases such as influenza and the common cold. Diseases
excluded from these models are, for example, host-vector and parasitic infections al
though they have some features in common. Modifications of epidemic models can
also be used in applications in the social sciences, modelling for example the spread of
knowledge or rumours (see e.g. Daley and Kendall, 1965, and Maki and Thompson,
1973). In this context being susceptible corresponds to being an ignorant (not having
some specific information) and infectious corresponds to being a spreader (spread
ing the information to other individuals). One possible rumour model is where the
spreader continues to spread the rum our forever which corresponds to the SI model
(e.g. Exercise 2.3).
Two main features make the modelling of infectious diseases different from other
types of disease. The first and perhaps most important reason is that strong depen
dencies are naturally present: whether or not an individual becomes infected depends
strongly on the status of other individuals in its vicinity. In Section 1.2 we shall
see how this complicates the stochastic analysis, even for a very small group of in
dividuals. For non-transmittable diseases this is usually not the case. Occurences of
such diseases are usually modelled using survival analysis, in which hazard functions,
describing the age-dependent risk for an individual to fall ill, are specified; propor
tional hazards (Cox, 1972) is an important example. These hazards may be specific
to each individual and even contain a random parameter which may be correlated
between related individuals; as in frailty models (e.g. Hougaard, 1995). Even then the
disease-times for different individuals are defined to be independent given the hazard
functions, contrary to the case for infectious diseases. A thorough analysis of such
models and their statistical analysis, including many examples, is given in Andersen
et al. (1993).
The second feature, which affects the statistical analysis, is that most often the
epidemic process is only partly observed. For example it is rarely known by whom an
infected individual was infected, nor the time at which the individual was infected and
during what period she was infectious (here and in the sequel we denote individuals
by 'she'). Problems related to this property are treated in Part II of the lecture notes.
In the introductory chapter we present the simplest stochastic model, discuss